A generalized floating-point quantum representation of 2-D data and their applications

In the past few decades, quantum computation has become increasingly attractive due to its remarkable performance. However, few results of 2-D data and their transformation based on quantum computation have emerged in recent years. A generalized floating-point representation of 2-D data (QR2-DD) is proposed, which can represent a quantum data with arbitrary size by the element of \(p + q\) qubits. And then, we provide a method to convert a 3-D data into 2-D data, which leads to a reduction in the number of qubits for position. Based on QR2-DD, the quantum circuits of elementary transformation of 2-D data are designed, such as interchanging two-row module, multiplying a row by a constant module and adding two-row module. In order to verify the effectiveness of these circuits, an example is given below each module.

The authors express their gratitude to the anonymous referees for their kind suggestions and useful comments on the original manuscript, which resulted in this final version. This work is supported by the National Natural Science Foundation of China (No. 41771375).

Authors and Affiliations

1. School of Mathematics and Statistics, Henan University, Kaifeng, 475001, China

   Rui Zhang & Dayong Lu

2. School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang, 471000, China

   Meiyu Xu

Corresponding author

Correspondence to Meiyu Xu.

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Appendix 1: Review of quantum computation

The states of a quantum mechanical system are represented by ket vectors or simply kets, denoted as \(|\cdot \rangle \). For example, ket \(\varPsi \) is represented as \(|\varPsi \rangle \). The ket vector \(|\varPsi \rangle \) belongs to an abstract space, referred to as the state space. Furthermore, any quantum mechanical state can be expressed as a linear combination of the elements of the state space. Many of the operations involving bras (the conjugate transpose ket) and kets introduced below are analogous to operations with vectors in Euclidean space.

Just as a classical bit has a state—either 0 or 1—a qubit also has a state. Two possible states for a qubit are the states \(|0\rangle \) and \(|1\rangle \), which as you might guess correspond to states 0 and 1 for a classical bit [24]. The arbitrary state \(|\psi \rangle \) is described by a linear superposition of the computational basis states:

where \(\alpha \) and \(\beta \) are complex number and satisfy \(\mid \alpha \mid ^{2}+\mid \beta \mid ^{2}=1\).

To consider the multiple-qubit case, it is necessary to introduce the concept of \(tensor \ product\). Suppose V and W are complex vector spaces of dimensions m and n, respectively. The tensor product \(V\otimes W\) is an mn-dimensional vector space. The elements of \(V\otimes W\) are linear combinations of tensor products \(|v\rangle \otimes |w\rangle \) of elements \(|v\rangle \) of V and \(|w\rangle \) of W. We often use the abbreviated notations \(|v\rangle |w\rangle , |v,w\rangle \) or even \(|vw\rangle \) for the tensor product \(|v\rangle \otimes |w\rangle \). The n-fold tensor product \(|v\rangle \otimes |v\rangle \otimes \ldots \otimes |v\rangle \) of \(|v\rangle \) is abbreviated as \(|v\rangle ^{\otimes n}\). By definition, the tensor product satisfies the following basic properties:

Suppose \(|v\rangle \) and \(|w\rangle \) are vectors in V and W, and A and B are linear operators on V and W, respectively. Then, we can define a linear operator \(A\otimes B\) on \(V\otimes W\) by the equation:

Pure states in quantum mechanics play an extremely important role in the preparation of states in quantum information. Quantum pure states refer to states that can be described by a state vector (including superposition states) such as

A quantum gate is simply an operator that acts on qubits. Such operators will be represented by unitary matrices. Some of the basic gates and their corresponding matrices are shown in (37). The identity gate (I), Hadamard gate (H), NOT gate (X) and controlled NOT gate (CNOT) are well known and can be found in the basic reference [24].

Appendix 2: Floating-point format in IEEE-754

Compared to a fixed-point representation, floating-point representation offers great savings in number of qubits when the required range of values and/or relative precision is large. The IEEE (Institute of Electrical and Electronics Engineers) is an international organization that has designed specific binary formats for storing floating-point numbers. The IEEE defines three different formats with different precisions: 32-bit single-precision floating-point numbers, 64-bit double-precision floating-point numbers, 128-bit extended-precision floating-point numbers. Format of floating-point numbers in IEEE-754 is shown in Fig. 13.

Format of floating-point numbers in IEEE-754

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Numbers in this format are composed of the following three fields:

1. 1.

   A sign bit denoted by s: \(s\in {\{0,1\}}\). When \(s=0\), the floating-point numbers are positive; when \(s=1\), the floating-point numbers are negative;

2. 2.

   An encoded exponent field denoted by e;

3. 3.

   The fractional part of the number denoted by f. The fractional part must not be confused with the significand (significand: the component of a binary floating-point number that consists of an explicit or implicit leading bit to the left of its implied binary point and a fraction field to the right), which is 1 plus the fractional part. The leading 1 in the significand is implicit. When performing arithmetic with this format, the implicit bit is usually made explicit.

The factorization of floating-point numbers \(\nu \) is

The value of bias is determined by the format and the number of e. For example, if \(0<e<255\), then \(\nu =(-1)^s2^{e-127}(1.f)\) for a 32-bit single format number \(\nu \). If \(0<e<2047\), then \(\nu =(-1)^s2^{e-1023}(1.f)\) for a 64-bit double format number \(\nu \).

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